Question

A population of values has a normal distribution with

Answer 1

Using the normal distribution, we have that:

• For a single value, P(X < 79.1) = 0.5517.

• For the sample of n = 155, P(X < 79.1) = 0.9463.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

The mean and the standard deviation are given, respectively, by:

$\mu = 76.2, \sigma = 22.4$.

The probability is the p-value of Z when X = 79.1, hence:

$Z = \frac{X - \mu}{\sigma}$

Z = (79.1 - 76.2)/22.4

Z = 0.13

Z = 0.13 has a p-value of 0.5517.

Hence: P(X < 79.1) = 0.5517.

For the sample of 155, applying the Central Limit Theorem, the standard error is:

s = 22.4/sqrt(155) = 1.8

Hence:

$Z = \frac{X - \mu}{s}$

Z = (79.1 - 76.2)/1.8

Z = 1.61

Z = 1.61 has a p-value of 0.9463.

P(X < 79.1) = 0.9463.

More can be learned about the normal distribution at brainly.com/question/15181104

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